Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients

TL;DR

This paper investigates the existence and non-existence of transition wave solutions in Fisher-KPP equations with complex, heterogeneous coefficients, establishing conditions based on wave speed and coefficient periodicity.

Contribution

It provides new necessary and sufficient conditions for the existence of transition waves in space-time heterogeneous Fisher-KPP equations, extending previous results to more general coefficient structures.

Findings

Existence of transition waves for sufficiently large wave speeds.

Non-existence of waves for small wave speeds, without requiring spatial periodicity.

Extension of methods to non-periodic space-time coefficients.

Abstract

This paper is devoted to existence and non-existence results for generalized tran-sition waves solutions of space-time heterogeneous Fisher-KPP equations. When the coefficients of the equation are periodic in space but otherwise depend in a fairly gen-eral fashion on time, we prove that such waves exist as soon as their speed is sufficiently large in a sense. When this speed is too small, transition waves do not exist anymore, this result holds without assuming periodicity in space. These necessary and sufficient conditions are proved to be optimal when the coefficients are periodic both in space and time. Our method is quite robust and extends to general non-periodic space-time heterogeneous coefficients, showing that transition waves solutions of the nonlinear equation exist as soon as one can construct appropriate solutions of a given linearized equation.